A binomial distribution is based on a probability experiment with a fixed number of trials, with
each trial having only 2 possible outcomes. A multinomial distribution, which is a type of
discrete probability distribution is based on a probability experiment where each trial has
more than two outcomes. For instance a person may have a choice of what to watch on TV, a
drama, a horror flick, a romance, an action movie, or a documentary. There are four requirements
for a multinomial experiment. Number 1 is that there must be a fixed number of trials. Number
2 is that each trial has a specific number of outcomes. Number 3 is that each trial
must be independent of one another, and number 4 is that the probability of a particular
outcome remains the same. The formula for the multinomial distribution is probability of x
equals, n factorial, divided by x1 factorial, times x2 factorial, times x3 factorial, and so on, to
xk factorial. Times p1 to the x1, times p2 to the x2, and so on, to pk to the xk. And x1
plus x2, plus x3, and so on to xk, equals n, and p1 plus p2 plus p3 and so on to pk equals 1. Ok, so a
lot to unpack here, but it's actually easier than you may think. In this formula probability of x
consists of several events, E1, E2, E3, and so on, and the x's represent the number of times each
of the different events occur, so x1 represents the number of times event 1 or E1 will occur, and
the p's represent the probabilities of each of those events, and n is equal to the total number
of occurrences of those events. So if we had 3 events, E1, E2, and E3, and event 1 occurred 3
times, event 2 occurred 4 times and event 3 occurred once, x1 equals 3, x2 would equal 4 and x3
would equal 1, and n would be the total number of occurrences of those events, 3 plus 4, plus 1,
which equals 8, and this is the number of trials. Let's look at an example, let's say you gather data
about people going to a fitness center and found that 20% we're doing strength training, 40% we're
doing cardio, 15% we're doing yoga, and 25% were playing a sport. If 8 people walk into the
fitness center, what is the probability that 1 of them is there to do strength training, 4 of
them are there to do cardio, 1 of them is there to do yoga, and 2 of them are there to play a
sport. First let's unpack the question, we have 4 different activities, strength training, cardio, yoga,
and sports. So these activities are the different events, E1 through E4. In the formula the
x's represent the number of times each of the different events occur, so x1 represents the number
of times event 1 or E1 will occur, which in this case is strength training, and the question says
1 of them is there to do strength training, so our x1 value is 1. x2 represents event 2,
which is cardio, and the question says 4 of them are there to do cardio, so our x2 value is 4. x3
represents event 3, which is yoga, and the question says 1 of them is there to do yoga, so our x3
value is 1. And x4 represents event 4, which is to play a sport, and the question says 2 of them
are there to play a sport, so the x4 value is 2. Now that we have all of the x's assigned values
we can sum them up, and that totals 8, which is the n value in the formula, and the p values,
the probability values of each of the 4 different events was given in the question. For
event 1 strength training, the probability is 20% or 0.20, which equals p1, for event 2, cardio, the
probability is 0.40 which equals p2, for event 3, yoga, the probability is 0.15 which equals p3 and
for event 4, to play a sport, the probability is 0.25 which equals p4. Now we have all we need to
plug and calculate. So the probability of x, or the probability that 1 person is there for strength
training, 4 people are there for cardio, 1 person is there for yoga, and 2 people are there
to play a sport equals, 8 factorial, divided by 1 factorial, times 4 factorial, times 1 factorial,
times 2 factorial, times 0.20 raised to the 1, times 0.40 raised to the 4, times 0.15 raised to the 1, times
0.25 squared. Now we can calculate out, which I've done on the screen, and we get a probability of
0.040, or 4%. So to sum this up, there is a 4% probability
that if 8 people walk into a fitness center 1 of them are there for strength training,
4 of them are there for cardio, 1 of them is there for yoga, and 2 of them are there to play
a sport. All right my friends, I highly recommend you write out all the notations and put in all
the values for them, a lot of times that can make things a lot clearer. Hopefully you got
something out of this video, I do have more videos right there for you, till next time, I am outta here.